How to make an animation play in the blender game engine?
First you had the logic bricks on the wrong object. You needed the Action actuator on the Armature not the mesh object, because it is the armature that is animated. 

Your second problem was the mirror modifier (I applied it). The mirror modifier was causing the animation to not play in the BGE. Just another one of the differences between the game engine and blender.
In general you should not have any modifiers or transforms (scale or rotation) on your objects in the BGE.

Your working blend file.